Kinematic mappings of plane affinities

verfasst von: Herbert Hotje
Abstract: In 1911 W. Blaschke and J. Grnwald described the group ℬ of proper motions of the euclidean plane ℰ in the following way: Let (P, script G sign)be the real three-dimensional projective space, let ℰ̄ ⊂ P be an isomorphic image of ℰ, and let U ∈ script G sign such that ℰ̄ ∪ U is the projective closure of ℰ̄ in P. Then there is a bijection κ : ℬ → P′ := P \U called the kinematic mapping and an injective mapping ℰ̄ × ℰ̄ → script G sign; (u, v) → [u, v] called the kinematic line mapping such that [u, v] := {β ∈ P′; β(u) = v} where the operation is denned by conjugation. A principle of transference is valid by which statements on group operations of (ℬ, ℰ) correspond with statements on incidence in the trace geometry of P′. Following Rath (1988) I will show that a similar concept holds for the group of affinities of the real plane where (P, script G sign) is part of and spans the six-dimensional real projective space.
Organisationseinheit(en): Institut für Algebra, Zahlentheorie und Diskrete Mathematik
Typ: Artikel
Journal: Discrete mathematics
Band: 155
Seiten: 121-125
Anzahl der Seiten: 5
ISSN: 0012-365X
Publikationsdatum: 01.08.1996
Publikationsstatus: Veröffentlicht
Peer-reviewed: Ja
ASJC Scopus Sachgebiete: Theoretische Informatik, Diskrete Mathematik und Kombinatorik
Elektronische Version(en): https://doi.org/10.1016/0012-365X(94)00375-S (Zugang: Offen)

BibTeX

@article{bcb8e9dd100a444dad435acee73b8faa,
title = "Kinematic mappings of plane affinities",
abstract = "In 1911 W. Blaschke and J. Grnwald described the group ℬ of proper motions of the euclidean plane ℰ in the following way: Let (P, script G sign)be the real three-dimensional projective space, let {\=ℰ} ⊂ P be an isomorphic image of ℰ, and let U ∈ script G sign such that {\=ℰ} ∪ U is the projective closure of {\=ℰ} in P. Then there is a bijection κ : ℬ → P′ := P \U called the kinematic mapping and an injective mapping {\=ℰ} × {\=ℰ} → script G sign; (u, v) → [u, v] called the kinematic line mapping such that [u, v] := {β ∈ P′; β(u) = v} where the operation is denned by conjugation. A principle of transference is valid by which statements on group operations of (ℬ, ℰ) correspond with statements on incidence in the trace geometry of P′. Following Rath (1988) I will show that a similar concept holds for the group of affinities of the real plane where (P, script G sign) is part of and spans the six-dimensional real projective space.",
author = "Herbert Hotje",
year = "1996",
month = aug,
day = "1",
doi = "10.1016/0012-365X(94)00375-S",
language = "English",
volume = "155",
pages = "121--125",
journal = "Discrete mathematics",
issn = "0012-365X",
publisher = "Elsevier",
number = "1-3",
}